On the other hand, one observes for \(k \in [1,3]\) , that (REF ) holds. It is trivially obeyed
for \(k = 1,2\)  because of the post–Minkowskian structure [1]}, see also [2]},
and is implied for \(k = 3\)  by
\(\Gamma ^3 {\chi }^{\rm h, loc}_3 - \chi _3^{\rm Schw}~&=&~\nu \left[-\frac{\eta ^2}{3p_\infty } -\frac{47}{12} \eta ^4 p_\infty -\frac{313}{24} \eta ^6 p_\infty ^3 - \frac{749}{320} \eta ^8 p_\infty ^5 - \frac{7519}{4480} \eta ^{10} p_\infty ^7 \right]+ O(p_\infty ^9),\nonumber \\\) 
